 The last talk is by Byrne Goldrisch. It's a very important one at the conference. He's telling us about the Mathematics contributions of Craig Hineke. Of course, those contributions are too vast to manage in one talk, but he's giving us a sampler. Thank you very much. Thank you very much, Srinivasi, for the introduction. And also thank you very much for giving me the opportunity here to talk about Craig's work. Of course, it is a challenge, as he said, because there's so much of it, and to make a selection is difficult, so it really is just a sampler. I also want to say on a personal note I've met Craig for the first time over 41 years ago at a Snowden Winter Meeting of the AMS in Cincinnati. And then in the following fall, Craig joined Purdue University as a faculty member, where I was a postdoc at the time. And that's when we started working. And this was the beginning of a long-lasting collaboration and friendship. And I've always thought of Craig very much as an older brother, although he's not so much older than me. But it is, even if we don't meet for a long time, there's always an immediate connection mathematically and otherwise, so it's always great to be with him. Of course, Craig has had a tremendous influence on the field and the people working in the field. He has been a big supporter of young people and he has helped all of us through his work, of course, but also through the many questions he asked and the problems he shared very generously. And he certainly has changed the face of commutative algebra. There are lots of new developments where he was at the very beginning and initiated them or he built the ground floor on which other people then could keep constructing theorems and new theories. So this is a great contribution and we all have to be very grateful to him. Of course, it's sad that Mel cannot be here, even I think virtually he won't be here. On the other hand, this talk I'm giving is also very much inspired by a joint talk and a joint paper. We wrote Mel and I about Craig's work at the occasion of his 65th birthday. So, somehow he's there at least. So let me start. So again, this is a sampler, not more than a sampler. I'm talking about some themes of Craig's work. One of the themes is asymptotic behavior of ideals and that includes restrings, integral closures, symbolic powers, but also linkage, residual sections and similar things. I also talk about uniformity, which is a big theme in his work and according to him, it's a very big thing in commutative algebra, certainly it is a big theme in his work. I also talk about local chronology and R plus. I won't touch tight closure because Irena Swanson will talk about that on Thursday. So that's the work of Craig and unity and Craig and Mel about tight closure. So I'm sparing that out completely. Okay, so let's start then. So since Hilbert functions and multiplicities were introduced, the behavior of powers of ideals has been an important topic in commutative algebra. This has led to the study of restrings, associated graded rings and the relationship between powers and symbolic powers for instance. And Craig already in his thesis, written basically under the supervision of David Eisenbatt, came up with a very important notion in this area, that's a notion of de-sequences. So what's a de-sequence? Going the wrong way. What's a de-sequence? De-sequence is a vast generalization of a regular sequence. So let's take N elements A1 up to AN in a Nuthian ring, Nuthian is not so important here. And I, the ideal they generate, then these ideals form a de-sequence. If the following is true, well, what does regular sequence mean? It means that if one colons an element into the ideal generated by the previous elements, you would get the ideal generated by the previous elements. Now de-sequence is the same, but you have to also intersect with the ideal generated by the whole sequence. So this is a much weaker condition, of course. And it is satisfied in many instances, and that's one of the things Craig showed. And one of the applications of de-sequences is the following theorem, except the theorem doesn't want to come. Sometimes they don't, yeah, it's true. It's dark. Oh, mm-hmm, yeah. Oh, there it is, okay, no? That's it, that's it, okay. So that's a theorem that was proved independently by Craig and by Tito Valla. And it says that if you have an ideal generated by a de-sequence, then the natural map from the symmetric algebra onto the Ries algebra is an isomorphism, okay? And this is the first big result in this direction before this was known for regular sequences and so on, which is quite, quite, not so interesting. And why is that interesting? Well, it has for two reasons. First of all, if the symmetric algebra is a Ries algebra, then one has control over the associated primes of the symmetric algebra, which are usually a mess. So because the Ries algebra is torsion-free over other symmetric algebra is not in general. And on the other hand, for the Ries algebra, it has the benefit that one knows the defining equations of the Ries algebra because one knows the defining equations of the symmetric algebra. And that is a classical problem in elimination theory. So that's one of the advantages and this is already a very important result. Now, remember, regular sequences can be characterized in terms of the acyclicity of the Kozul complex. And there's a similar characterization for de-sequences. They can be characterized in terms of the acyclicity of another complex, which is called the M-complex. And that complex was introduced by Herzog, Siemens, and Vasconcelos. And what it is, is the following. I quickly want to mention the construction. So we start again with these elements in a new theorem ring. I see ideal degenerate. And then we look at the homology of the Kozul complex on these elements. So that's a graded model, if you want. And then one can construct a complex called the approximation complex. And what it is, is you take H and tensor it with a polynomial in N new variables, as many variables as there are generators of the ideal. And the differential is the Kozul differential, but not the Kozul differential on the A, is that would just be zero because I've taken the homology of the Kozul complex. Rather, one takes the Kozul differential on the T's on these new variables. And that's a bounded complex. And it turns out that it's zero's homology is always a symmetric algebra of i mod i squared, which maps onto the associated graded rings. So that's sort of the same picture we had before the symmetric algebra mapping on the least ring, just the same picture but tensed with i mod i. So that's what this gives us. And it turns out, and that was proof by Herzog-Cymus Vosconcelos, that if a ring is local with infinite residue field, then the acyclicity of the M complex is equivalent to the fact that i is generated by a d sequence. So this is a characterization, not of d sequence, but of ideals generated by d sequence. That's the difference, of course. So that's the homological characterization of d sequences and that connects this notion of d sequence to homological algebra, which can be used there effectively. And for instance, they prove also that if as a kormikoli ring and the height of i is positive and two important conditions are satisfied and they will appear over and over again in this talk. The first is the strong kormikoliness. i is strongly kormikoli if the causal homology is a kormikoli model, automatically a maximal kormikoli arm model. It's called strongly kormikoli because arm or i is h0. So strongly kormikoli in particular implies that arm or i is kormikoli, but it's much stronger, of course. In the second condition, that's a simpler condition. It's a condition on the local number of generators of the ideal. It simply says that locally at each prime, the number of generators of the ideal is at most as big as the dimension of the ambient ring. So that's much weaker than being a complete intersection locally. This is a sliding condition and that can be easily checked in terms of height or fitting ideals. And what this does for you is that the strongly kormikoliness tells you that the M-complex is a bounded complex of maximal kormikoli models. And this other condition tells you that locally, since the number of generators is small, the M-complex can be replaced by a complex of a very short length. And these two facts together with the acyclicity lemma imply immediately that the M-complex is acyclic. So this is a sufficient condition for acyclicity of the M-complex. And once you have the M-complex as acyclic, then one can show in general, if the M-complex is acyclic, then this map up there from the symmetric algebra of I mod I squared onto the associated gradient ring is actually an isomorphism. And that isomorphism, again, by work of Valar lifts to an isomorphism between the symmetric algebra and the associated gradient ring. That's one way to see it. So acyclicity implies this isomorphism. And then the kormikoliness of this associated gradient ring, for instance, follows because now the M-complex is acyclic complex and all the models are maximal kormikoli models. Therefore, by the depth lemma, this associated gradient ring is kormikoli. And then, for instance, by work of Chungiketa, the restring also has to be kormikoli. So that's one way to say it. So this is really all one wants in some sense. It's very important to know that these algebras, the restring and the associated gradient ring are kormikoli. And it's also very important to know that the restring is isomorphic to the symmetric algebra. So this is somehow the desired result. Of course, there are drawbacks I'll talk about it a little bit later, okay? Now, Craig proved similar results, not using this homological machinery, but using what he calls kormikoli de-sequences. But he came up with very similar results with different methods, okay? Now, the question is, the condition here, as I said, there are two conditions. One is sort of harmless, this condition on the local number of generalists. The other is a subtle condition, strong kormikolness. And then the condition is, which ideas are strongly kormikoli? How do you check? Do you have good examples? And this is where Craig comes in again. And this is where linkage comes in. So when are ideas strongly kormikoli? And this needs to the notion of linkage. Linkage or liaison has been used since the 19th century as a method for classifying projective varieties in particular curves in P3, and then names are Max Nöter, Kayleigh, Hylphen, Gaeta, and many others. And here's the definition. Suppose R is a kormikoli ring and I and K are proper ideals. Then these ideals are linked, and that's the symbol one uses. If there exists a regular sequence, which has the property that K is the colon ideal, alpha colon I, and for symmetry, I is the colon ideal, alpha colon K. And from this definition, it follows immediately that alpha is in the intersection of I and K, and also that I and K both are mixed of the same co-dimension G, okay? So, and this relation is symmetric, but it's not reflexive or transitive, and therefore one passes to what's called the linkage class, the equivalence relation generated by this. Two ideals are in the same linkage class, or the same even linkage class. If for some N or some even N, I can be linked to K by a finite sequence of links, N of them, okay? And one says that's a particularly interesting case that I is leachy if it is in the linkage class of a complete intersection. So the leachy ideals are those which you can get from a complete intersection via iterated linkage, okay? And that's an important class because these ideals can be arbitrarily complicated. On the other hand, there are somewhat well behaved. They still remember that they come from a complete intersection a long time ago. So that's the advantage. And here are some examples. Perfect ideals of high two are leachy. This is basically the content of some of this classical work. Of course, they were talking about curves in P3, but and then the condition is that these curves are automatically called if and only if they're leachy. The same works for perfect ideals which are Gornstein of high three. And but it also works for other types of ideals. For instance, this ideal here, which is generated by the minus maximum minus of a generic matrix together with the entries of this product. So a vector times a matrix. So this is a baby case of a variety of complexes. And this is also linkage. This ideal is also the defining ideal by the way of the associated credit ring of a generic perfect ideal of high two. And those are examples of linkage. Leach ideas are many more and constructions how to get new leachy ideals from old ones. But here's the theorem of unity. This is amazing and they're interesting result. And it's really one of the first results that prove really non-trivial properties of leachy ideals. And it says that if we are in a Cormier-Cawley ring and INKI deals in the same even linkage class, even is important, even is closer than odd K because each ideal is evenly linked to itself but it's only linked to something totally different possibly. So then I is strongly Cormier-Cawley if and only if K is. So strong Cormier-Cawley is an invariant of the even linkage class. And in particular, since every completender section is strongly Cormier-Cawley and every completender section is linked to itself so even odd doesn't make a difference. It follows that every leachy ideal is strongly Cormier-Cawley. So all these ideals I mentioned before they all satisfy the assumptions in that theorem about associated graded rings and tree strings. So that's a very, very beautiful result and interesting result. Then in joint work we went on to study leachy ideals and not only leachy ideals to study other properties that are invariant under linkage in some sense. And one of those properties are the graded shifts in a minimal free resolution. So now take a homogeneous idea, which is perfect, call it I prime, in a polynomial ring, look at the minimal homogeneous resolution of this idea. Sorry, of this, what happened? Wrong button, okay, as usual. We look at the minimal free resolution of this idea. The resolution before the perfection has length G minus one where G is the co-dimension of the ideal and now localized at normal G is maximum ideal. And the statement is if the maximum of the last shift is at most the length of the resolution times the minimum of the first shift. So in other words, if the shifts don't grow very fast then for every ideal in the linkage class of I, even after localizing, and these ideals of course need not be coming from homogeneous ideal anymore. Any ideal, even in this localized ring one can say that the number of generators of this new ideal cannot be smaller than the number of generals of I and the type of R mod K cannot be smaller than the type of R mod I. So in other words, no matter what you do, you can never improve the ideal by even linkage. If this particular condition is satisfied, in particular, I cannot be leechy because if that condition is satisfied then I would have to be a completeness section but completeness sections are ruled out by these numerical conditions. So such an idea can never be leechy. One should think of this as a negative result in some sense a criteria, a necessary condition for being leechy. So one can use that to show something is not leechy and one can use it also to distinguish linkage classes. And in particular, one can get from this quite easily. The following, where we had seen already that there's only one linkage class of arithmetically corn macaury curves in P3 and this was known classically not in this language but this was known in the 19th century already. And also of arithmetically corn steam curves in P4 but using this previous result, one can easily show that there are infinitely many even smooth arithmetically corn macaury curves in P4 or arithmetically corn steam curves in P5 that belong to different linkage classes. So the behavior is completely wild if you step up one dimension and in fact these various linked curves they can even be obtained from each other by a linear automorphism of the projective space. So this way linkage is very, very fine. Because linkage is very fine, one wants to look at a more general notion and that's a notion of residual intersection, okay? And residual intersection is similar to linkage except that the two so to speak linked ideals need not have the same height. So you can have different ideals connected to each other. Otherwise the definition is similar. So our again is a new theorem ring, I is an ideal of height G, S is an integer at least G and then one considers a proper ideal K and such a proper ideal is an S, the residual intersection needs to keep track of this S of I if K is a colon ideal, J colon I for some S generated ideal J inside I and the height is at least S. So the crucial point here, the crucial condition that makes this somewhat strong and rigid is that the number of generators of the ideal one colons into is less or equal to the height of the colon, okay? You may be tempted to think that S is the largest possible height of this colon but that's not true. There is no cold altitude theorem for colon ideals. In fact, this height can be arbitrarily large. And it's actually interesting when the height cannot be arbitrarily large or what the height is because that's related to integral dependence of ideals is also related to dimensions of secant varieties and so on. So there are various interesting connections to that. Now assume for instance that I is Gornstein and I is unmixed that it's easy to see that G is cylinder section. So the case where S is smallest possible and M is G, the height of the ideal is simply linkage. So this is a vast generalization of linkage. But notice the situation is very different because if two ideals are linked and if you are in a Gornstein ring, then if Armot I is comicale, then also the link is comicale. That's a result proved by Peskin and Spiro. So in modern algebraic language, they reworked the classical theory of linkage in modern algebraic language and that's what they proved. They also observed that this is not true if the ambient ring is only comicale. So it doesn't work in a comicale ring. And then the crucial observation of Craig was that if you're in a comicale ring and you strengthen the condition of being comicale to being strongly comicale, then indeed the link is comicale, okay? That's a very crucial observation because of the following. It allowed him to prove a theorem which is a very surprising theorem because in general it's a theorem about the singular section. Of course in general the singular sections no matter if your ideal is comicale the singular sections need not be comicale anymore. They can be badly mixed and all kinds of strange things can happen. But nevertheless he can show the following. If we are in a comicale ring, i is an ideal and s is an s singular section so that familiar conditions, first of all, i is strongly comicale and second this g condition on the number of generators except now it's not g infinity. It's only what's called gs which means it's like g infinity but you don't require it for all primates. You only require this for primates up to height s minus one. So in particular if s is at most the dimension of the ring you'll never say anything about the global number of generators of the ideal. So this is weaker condition what we have seen before but then he shows that in this case indeed the height of k cannot be bigger than s so this is a cool intersection, cool altitude theorem for colon ideals and furthermore r mode k is comicale. So this is a rather surprising result and the way this observation comes in is one can reduce residual sections to links but on links on residual sections of smaller co-dimension but those will not be Gornstein anymore even if the original ambient ring is Gornstein but by induction they are comicale so one has to do linkage in comicale rings and what saves the day is the strong comicale assumption because even if you're only in a comicale ring the link will remain comicale if the original ideal is stronger than comicale namely strongly comicale and that was a key observation. There was a previous result by Art and Nagata but it was not correct and Greg observed that causal homology comes in or comes to help here. Okay so let's go back to these rings and establish a connection between these rings and residual sections. So let's look at an ideal, these are without any assumptions really. Let's look at this ideal. I is generated by n elements a1 up to an and as an ideal in R and R is of course a degree zero part of the symmetric algebra but you can think of I also as a degree one part of the symmetric algebra and then I denote the generators by a1 prime up to an prime and then one can form what's called the extended symmetric algebra and it's called SIMI, a joint T inverse this makes no sense it's just a symbol you can think of it as a T inverse but what it is it's just a symmetric algebra a joint available modulo all the relations that tell you that you can get from the a primes to the a's by multiplying with u so this is why the u works like a T inverse and this extended symmetric algebra maps on to the extended Ries ring which is R a joint I T comma T inverse and now the T inverse really does make sense in the Laurent polynomial ring and the u goes to T inverse there's always this surjection and now Craig and that's the crucial connection to the serial intersections proved that if I has positive height and is G infinity so this justice condition on the local number of genus no depth condition on causal homology or so or anything like this then this extended symmetric algebra is defined by an ideal K so we can write it as a polynomial ring modulo an ideal K which is an N residual section of the original ideal not quite of the original ideal together with this web so the defining ideal of the extended symmetric algebra is a residual intersection of the original ideal this gives a very tight connection between the serial intersections and Ries rings or extended symmetric algebras and so on and let me give an application of this which follows immediately from his work now assume again I score McCorley the height is positive and now throw in the stronger condition that I strongly call McCorley and G infinity then under these assumptions because of G infinity we know that the extended symmetric algebra is defined by a residual section by the previous theorem of Craig because we have strongly called McCorley and G infinity we know that residual sections are called McCorley so therefore the extended symmetric algebra is called McCorley since the extended algebra is called McCorley one can quite easily also prove that it is equal to the Ries algebra because one is enough to check this equality at the source at primes if one has called McCorley and one has control over source at primes so one gets this equality but then also the source at credit ring is called McCorley because simply the extended three string modulo principle ideal and from that again one also gets that the symmetric algebra is isomorphic to the Ries algebra and it is called McCorley okay this isomorphism of course the first isomorphism immediately implies the second isomorphism and the call McCorley simply comes again if you want from Chung Ikeda since the source at credit ring is called McCorley the Ries ring is called McCorley in this case so one can completely recover this result by Herzog-Cymus Vasconcelos simply from the cylinder sections and this sets up this connection between Ries rings and the cylinder sections in this setting where we have G infinity if we don't have G infinity this connection is less apparent but still is there and is used a lot so and that's a very important insight that came from Craig's work now so we've talked about Ries rings with the G infinity condition of the ideal of course the problem is that if the ring is local then G infinity implies in particular that the number of generators is bounded above by the dimension of the ring so that's of course a strong restriction it's a restriction on the number of generators that's something we don't want of course without that restriction you cannot expect that the symmetric algebra and the Ries algebra are isomorphic but one still wants that perhaps the Ries algebra is McCorley without being isomorphic to the symmetric algebra and Greg together with Sam Huckabar were the first to treat systematically the case of ideals that have arbitrary number of generators and are not integral over parameter ideals so that's a very huge step into a new direction of considering algebraic properties of Ries rings and as I said the problem is that the number of generators of the ideal I is too large so what you do you pass to an ideal which has smaller number of generators but still is very connected to the ideal and that's the minimal reduction of an ideal so we have to talk about reductions and integral dependence of ideals so one says that I is integral over J J is a sub ideal of I we are in a new theorem local ring with infinite residue field I is integral over J so I currently J is a reduction of I in some sense that's just different language if and only if the induced inclusion of Ries rings is an integral extension of rings in the usual sense and if you read this fact degree by degree you get this condition namely that the R plus first power of the larger ideal is obtained from the Rth power of the larger ideal times one power of the smaller ideal eventually for all R and the smallest such R where this works is called the reduction number of I with respect to J and it is written in this way so in some sense reduction is a simplification of the ideal and the reduction number measures how far the two ideals are apart from each other and then one wants to look at minimal reductions a reduction is called minimal if it is minimal with respect to inclusion they always exist by work of Northcott and Trees and another definition the analytic spread actually came up in Dale's talk already analytic spread even of filtration this is just analytic spread of Iatic filtration the analytic spread of I is the minimal number of generators of J for any minimal reduction of I so minimal reductions are highly non-unique but they all share the same minimal number of generators and that's because this invariant can also be defined as the Krull dimension of the special fiber ring the Ries ring tensed with K so that's a special fiber over the close point inside the blow up the brooch of the Ries ring the blow up of spec R along V of I and from this definition it's clear that the analytic spread is bounded above first of all by the dimension of R but also by the minimum number of genitals of I and it's bounded below by the height of I because I and the minimal reduction they have the same radical so therefore the height of I is the height of J and L of I is the number of genitals of J so simply use Krull's attitude here but now here we are, now we have a new ideal J which satisfies this condition we didn't have before namely that the number of generators is bounded above by the dimension of the ring so if I replace I by J then the number of generators is bounded above by the dimension of the ring R so one could maybe hope one has some information about the Ries ring of J and then goes back to R if reduction number is small enough and in fact Greg and Sam were able to implement that so write G for the height of I and L for the analytic spread and if the analytic spread is as I said is at least G if it's equal to G the case is kind of trivial, well understood but if not the first case to consider is the case where the difference between L and G is at most one or equal to one that's called analytic deviation one and in that case they prove what one would like to have again the assumptions are R is a Cormac-Corley ring let me go to the next, sorry R is a Cormac-Corley ring R mod I is almost Cormac-Corley means at either the depth is the dimension or at most one worse the height is at least one I is a completeness section local encode dimension G plus one and L that's the crucial assumption is at most G plus one and then the reduction number of I with respect to some minimal reduction is at most one and from that they get that the restring and the associated Cormac-Corley so but notice there's no condition on the number of generators of the LDI and they have a similar result for G plus two one has to strengthen the conditions a little bit but that's what they prove now I should say that meanwhile this has this this generate a lot of activities in the first half of the 90s many people worked on this and the eventually the case of Arbiter analytics spread was solved and the pools were different and so on but the main point is that again Greg and in this case with Sam they were at the foundation of this that they were the first ones who started with and maybe people never would have picked up this or they had to pick up this I mean their proof is very very complicated but things got much simpler as time went on but that's not the point I mean they were the point the ones that started this and of course in general one has to impose conditions on the reduction number one has to also impose some conditions on depths of powers or causal homology one just doesn't get away with these basic conditions but still this is the way it went in both case also interesting to observe that j-core and i is a residual intersection so there is again this connection still to residual intersection although the proofs don't go quite directly in terms of translation from one to the other okay let me now also talk briefly about the core I had mentioned that in general minimal reductions are highly non-unique and to remedy this one looks at the intersection of all reductions that's called the core of an ideal so we are in a Northean local ring infant residue field i is an ideal then the core of i is the intersection of all reductions is enough to look at all minimal reductions and this was introduced by Judy Sally and Reese, Reese and Sally and it also comes up naturally in the Pryon-San's Coda Theorem I'll get back to this later if i is regular then i raised to the power l of i the analytical spread is actually contained in the core so that's a lower bound for the core but in general the core is an up-rheor infinite intersection and it's very hard to compute or understand and the first again to find a formula very explicit formula for the core where Greg together with Irina Swanson this is now in a two-dimensional regular ring i is an integrally closed ideal so we have the Zariski theory of integrally closed ideals in regular rings and then they prove that the core of i is i times the second fitting ideal or j squared core i so the core which is an infinite up-rheor intersection of ideals well in this case not but in general it could be an infinite intersection then can be expressed in terms of a single reduction j is a single reduction j is a one minimal reduction again this was vastly generalized but this is really the ground floor so that's a very very it's also very different not easy proof at all so now let me pass to the other topic namely uniformity so uniformity in netherian rings so here's a quote from Greg Yonecki behind the obvious fineteness condition in netherian rings there lie many deeper and hidden types of fineteness which come to light in terms of uniform behavior so one way to think of uniform behavior is that it's that uniform means uniform for every ideal i in a given netherian ring R so I fix the ring R but I want uniform behavior for all ideals or study what uniform properties to all ideals have and a typical example and one that also inspired Greg as far as I understand is the following suppose we are in prime characteristic then a test element from tight closure theory has such a uniform property namely it multiplies the tight closure of any ideal into the ideal so in other words it's a uniform annihilator of the module i star modulo i regardless of what i is and that uniformity plays a very important role in tight closure theory here I will focus on Greg's work on uniformity related to the Artin-Ries theorem interclosure filtrations and symbolic power filtrations so let me explain this what is the Pryonson-Skoda theorem where here's the Pryonson-Skoda theorem proved by Pryonson-Skoda with analytic methods and the first algebraic proof and the general algebraic formulation is due to Liebman and Sartre so assume as a regular domain of dimension D finite then what the Pryonson-Skoda theorem does it compares the integral closure filtration with the iatic filtration in a uniform way and in a very tight way and what it says is that the integral closure of i n is contained in i n minus a constant namely minus D plus one so that's I'm adding or subtracting an integer and I get this contained so there's a very tight connection between the two filtrations but the point also is that this constant I'm adding or subtracting is independent of the i d i so it's a uniform constant so that's a uniform behavior now one way to think of this and that's the way they proved it actually is that this is simply a statement about the conductor of the extended res algebra it simply says that the element T to the one minus D is in the conductor of the extended res algebra and then the way the proof goes is when we just have to approximate the conductor from below one has to show that this particular element is in there and the way they do that is that they show very generally that Jacobian ideal is in the conductor and that's of course generalizing very classical results that's unknown algebraic number theory also in four rings for f n k-algebras by Amy Nerda and so on this was all known but of course they need this in much greater generality and that's what they prove and that's how the proof goes now one cannot expect this to be true if the ring is not regular because for instance if D is one in the one dimensional case this inclusion would say that all ideals are integrally closed but then the ring is regular so that doesn't work however in the one dimensional case you also have that you have the conductor ideal say if you're in a local case and then any non-zero ideal say if you're in a domain every non-zero ideal contains if you have an M primary ideal or an ideal which is not the whole ring then a power of that ideal is always contained in the conductor of the ring and from that alone you would get some kind of a uniform condition like this it wouldn't be the D minus one or one minus D but it would still be uniform so one could expect possibly that a uniform result of this type works and but of course before that one should see why in general such a result at all could work even if I fix the ideal I and it doesn't work in general that's a result of these but it does work if the ring is analytically unrhymified so if the ring is analytically unrhymified then for every ideal there exists a K so that I n bar is contained in I to the n minus K so that's simply the fact that in this case the integral closure of the extended ring is finitely generated over the extended ring and that's guaranteed by these if R is analytically unrhymified analytically unrhymified means the ring is local and the completion is reduced but he also showed that the converse holds if this inclusion holds for a single and primary ideal then the ring has to be analytically unrhymified so that's the weakest condition you can impose K but of course now the next step is assuming analytically unrhymified can you choose this K uniformly K and Greg calls this uniform bryonsonskoda uniform bryonsonskoda holds in R if that K can be chosen uniformly and he conjectures that uniform bryonsonskoda holds in any excellent ring of finite dimension so and that's a very vast conjecture and he essentially proved it so now this is related there's another uniformity type of result that's the uniform art in Ristium so let me quickly explain this so N in M is an inclusion of Mithian R modules then the art in Ristium says that there exists a K so that this intersection property holds so simply that says if I look at the iadic filtration on M and I restrict it to the submodel it's of course not an iadic filtration anymore but it's still at least it's i stable so that's the art in Ristium and which is used all over the place now what about, sorry this is a typo I shouldn't have said it uniform art in Rist of course what one wants is that the K doesn't depend on I but one wants not this equality one just wants the inclusion that I intersect, I N intersected, I M intersected N is contained in I K M intersected with N the equality one doesn't agree that's the containment that the left is in the right without the, this is not sorry what one wants is I, I should write sorry what one wants is that the left hand side is an I to the N minus K N, N that's it which of course holds so so that's the crucial content that's the one that's used and that's uniform art in Rist and here's the theorem which and then Craig also conjectured that uniform art in Rist holds for any excellent ring of finite cold dimension and he essentially proved this so that's the theorem, amazing theorem it says that uniform, Brinus von Skoda and uniform art in Rist hold in any reduced new theorem ring are satisfying either of these conditions either as essential finite type over an excellent local ring or over Z or R has prime characteristic and is F finite and the reduced and excellent condition is only used for the Brinus von Skoda theorem not for the art in Rist, uniform art in Rist and notice what separates this from the completely general case is that in the first bullet point the ring is supposed to be local if that wouldn't be there that would be the general case but it's very close to being general so let me quickly outline the proof just to give an idea of these things are related the proof of these two uniformity properties Brinus von Skoda and art in Rist are intertwined and use two other uniform properties and the first one I just call them uniform conductor elements because that should really remind you of the proof of the Brinus von Skoda theorem because it relates to conductors and what you want in the end is to find uniform conductor elements for extended Rist rings regardless of the ideal and that simply means there exists a K and an element C in R naught R naught this is a notation from tight closure theory it's a set of all elements which are not in any minimal prime so that fixed C times the integral closure of I to the n is contained in I to the n minus K for every ideal I again this should be uniform or equivalently there exists that's essentially what it says there exists an element in the conductor of the extended Rist ring of degree minus K and except one doesn't take the conductor with respect to the integral closure one just take the integral closure inside the Laurent polynorm ring because I is not assumed to be normal so that's this uniform conductor condition but the point is that this should hold for every ideal now there's another condition another uniformity condition that's uniform annihilation of homology and it says there exists an element D again in R naught that annihilates the positive degree homology for every bounded complex of finite free R models that satisfy the Buxbaum-Eisenbach acyclicity conditions with grade replaced by height and so such complexes of course are not acyclic because in the Buxbaum-Eisenbach criteria you need grade not height but those are the complexes that would be acyclic if the ring were cormacoli so this takes care of rings that are not cormacoli and well with that the proof Greg's proof works like this if you have these conditions A and B not just for R but R modulo P for every primate ideal then he shows that uniform Artyn Ries holds for the ring so the uniform Artyn Ries for the ring follows from these two conditions where the first condition A this uniform conductor condition very much smells like the Artyn Ries theorem smells like the Brinozon-Skoda theorem so that's the connection in some sense Brinozon-Skoda sort of implies Artyn Ries I mean very roughly speaking and the proof is a induction on the dimension of M modulo N it's rather involved but that's the connection and the other connection so how do we get uniform Brinozon-Skoda if we have condition A and uniform Artyn Ries then uniform Brinozon-Skoda holds and this is how everything is connected for uniformity properties are very much connected and that proof at least I can show this really very clear and easy so for every ideal we have that C times the integral closure of IN is in I to the N minus K that's simply by part A but this ideal is also in the principle ideal generated by C so therefore I can intersect with C R and this works by A and for every I without any restriction I uniformly but then I can use this is now a condition that looks like like uniform Artyn Ries this version here and we apply this to the case where M is the ring and N is the ideal generated by C and then we get that this intersection is in C times I to the N minus K minus some K prime and by uniform Artyn Ries and now cancel the C season on zeal divisor and you get uniform Brinozon-Skoda so this condition A together with uniform Artyn Ries implies uniform Brinozon-Skoda Okay. That's what I want to say about uniformity let me pass to symbolic powers again I've come back to uniformity at the end of that but of course symbolic powers that's also a case where we want to look at symbolic on uniform properties so either Nithyan ring is an ideal W is the complement of the union of the associated primes and then the end symbolic power is the extension of the Nth power to the localization contracted to the ring now with this definition the first symbolic power is the ideal itself of course there's a difference where one takes associate primes or minimum primes and it contains always the Nth power now symbolic powers have appeared since long in commutative algebra for instance they played a crucial role in Kuhl's original proof of the Kuhl altitude theorem they have a geometric meaning for instance if R is a polynomial over a field and I is radical then the Nth symbolic power is the intersection of M to the N where M is any maximum ideal in V of I and this of course is a set of all polynomials that vanish to order N at each closed point in general though symbolic powers are difficult to compute difficult to understand also difficult to understand how they relate to powers and of course the simplest question not the simplest but the most basic question one could ask the best one could hope for is when are symbolic powers equal to powers and there is a nice observation by Craig that if this whole say suppose they are no embedded associate primes then this implies that locally at every non-minimal prime the analytic spread is strictly less than the height of the prime so that means the analytic spread cannot be maximum the analytic spread is always less or equal to the height of the prime but the equality has to be strict so that's a very strong condition and interestingly and that's together with David Eisen but the converse holds if the associate graded thing is chronic quality so what this says is that if one knows that the associate grade drink is chronic quality then this condition on the analytic spread alone which is easy to check implies that the powers and the symbolic powers are the same and since Craig proved a lot of results about chronic quality of associate grade drinks he there but also proved a lot of results about the quality of symbolic powers and powers that's just this necessary condition you have to check and if it's satisfied the powers and the symbolic powers are the same now if Eisen excellent domain then he also observed that if one simply has this strong condition on the symbolic on the analytic spread alone one can conclude that the symbolic res ring which is the direct sum of all the symbolic powers is a new theorem ring so it's not necessarily equal to the res ring but it's at least a new theorem ring in fact it's even a new theorem model over the basic that's even stronger okay so that's one condition one gets and he in general he has a lot of work on when this symbolic res ring is called Macaulay without this strong condition on the analytic spread and that is interesting because for instance if we are in a new theorem local ring and either one dimension prime ideal then the new theorem is of the symbolic res ring implies that the ideal is a saturated completeness section and we had talks about this so for some time this was a very active subject and Craig contributed greatly to that now so that's this question when the symbolic algebras new theorem in general makes sense and what I said it implies saturated completeness section in the one dimension case now in general okay powers and symbolic powers are not the same the best one could hope for is a comparison and perhaps uniform comparison between ordinary and symbolic powers and there was a result by Ein, Larsersfeld and Smith which gave a very tight uniform comparison between powers and symbolic powers for reduced ideals in regular rings essentially a finite type over a field of characteristic zero and this result and they used asymptotic multiply ideals and this result was greatly generalized by Hawks and Unicard to the case of any regular ring containing a field for instance and they also treat the case where the ring is not regular and here's the statement so I as a new theorem ring containing a field I is an ideal and H is the maximal height of an associated prime of I if as regular then this symbolic power of I to the H and is contained in I to the N so again a very tight bound so it says in particular that the symbolic topology and the ideal addict topology are linearly equivalent but that this factor H is actually uniform it's only can be bounded for instance by the dimension of the ring if the dimension of the ring is finite and of course one has to take at least something like a linear function where the coefficient of N is not one because if as before you would just take N plus a constant that wouldn't work because it would imply the symbolic ring to be integral over the ring and then this would that's a very special case so in general one cannot expect that so this H has to be at least so N plus a constant would not work so it has to take at least a constant times N so that's the least of it that's the best one could expect for and if the ring is not necessarily regular but you're on a setting where you can talk about a Jacobian ideal then if you multiply the left hand side with the N plus first power of that Jacobian ideal you still have the containment and the proof is via reduction to prime characteristic and there they show that the Nth power of the Jacobian ideal times that symbolic power is contained not in I to the N but in the tight closure of I to the N and that immediately gives the case of the regular ring because in the regular ring the Jacobian ideal is the unit ideal and the tight closure of any ideal is the ideal so that does it and for two one uses that the Jacobian ideal can be generated by completely stable test elements which means that if we multiply this inclusion by J then the right hand side is actually contained in I to the N and that finishes the proof. Well maybe I skip this and go immediately to the uniformity questions so again this is the simplest version of the Unique Rockstar theorem as a regular ring containing a field of finite and the ring has dimension D which is finite then I to the symbolic power D N is contained and I N for every I so that's the uniformity so this D this coefficient only depends on the ring not on the ideal that's simply because whatever this H was in the previous theorem is always bounded by the dimension of the ring and the question then is can one generalize this to rings which are not regular and just forgetting even about uniformity one could ask is there such a constant K for a given ideal so that this inclusion holds so that is sort of the analogue of the Ries theorem but now for the symbolic power filtrations and indeed that's true by result of Swanson for instance if I is normal and excellent then that can be done. And then the question of course is when can one choose this K uniformly and this is a theorem of Unique cuts in Validasti and they prove indeed one can and the main condition here is that the ring has an isolated singularity that's the restricting condition so R is analytically irreducible and with the local in it has an isolated singularity and then two conditions that should remind you of something I said before R is essentially of type finite type over a field of characteristic zero or R has prime characteristic and is a finite and then indeed there exists a uniform K so that the KN symbolic power is contained the Nth power for every prime ideal at least. So the proof uses and that's why these assumptions look familiar uses the two uniformity results I mentioned before the uniform Priyansar Skoda and the uniform Artin Ries which hold in this case by the previous theorems but it also requires new uniform versions of Chevrolet's theorem and many other things and this is still an open problem in general but Craig and lately with Dan Katz they have done work they considered for instance a finite abelian extensions of regular rings and under suitable additional conditions there they can still do that. I think I had some more things but maybe it's better to stop at this point. The questions or comments? This one? Where the main thing is that you just need basically that the integral closure is a finite model that actually is a little bit overkill but that's what you need. Any other questions or comments from any of the people on Zoom? Just thank you, Berent, for such a nice talk. I shouldn't say nice since it's about my work. Sure, that's why it's nice. Oh, there you go. Good, good, good to see you. That was one of my great pleasures was working with Berent for so many years and as he said it's like we can pick up where we left off any time. It's a wonderful thing to have a collaborator. So thank you, Berent. Thank you, Greg, thank you. So many years of pleasurable work. Let's thank Bernt and Greg also for the comment.